Level 4 Puzzle

354 and Level 4

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354 is even and thus divisible by 2, so 354 is a composite number. It is also made from three consecutive numbers so it is divisible by 3. Scroll down past the puzzle to see its factors.

354 Puzzle

Print the puzzles or type the factors on this excel file: 12 Factors 2015-01-12

  • 354 is a composite number.
  • Prime factorization: 354 = 2 x 3 x 59
  • The exponents in the prime factorization are 1, 1, and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1)(1 + 1) = 2 x 2 x 2 = 8. Therefore 354 has exactly 8 factors.
  • Factors of 354: 1, 2, 3, 6, 59, 118, 177, 354
  • Factor pairs: 354 = 1 x 354, 2 x 177, 3 x 118, or 6 x 59
  • 354 has no square factors that allow its square root to be simplified. √354 ≈ 18.815

354 Logic

346 and Level 4

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Quick Prime/Composite Number Test: 346 is even and thus divisible by 2, so 346 is a composite number.

346 Puzzle

Print the puzzles or type the factors on this excel file: 10 Factors 2015-01-05

  • 346 is a composite number.
  • Prime factorization: 346 = 2 x 173
  • The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1) = 2 x 2 = 4. Therefore 346 has exactly 4 factors.
  • Factors of 346: 1, 2, 173, 346
  • Factor pairs: 346 = 1 x 346 or 2 x 173
  • 346 has no square factors that allow its square root to be simplified. √346 ≈ 18.601

346 Logic

326 Tiny Christmas Factor Tree

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  • 326 is a composite number.
  • Prime factorization: 326 = 2 x 163
  • The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1) = 2 x 2 = 4. Therefore 326 has exactly 4 factors.
  • Factors of 326: 1, 2, 163, 326
  • Factor pairs: 326 = 1 x 326 or 2 x 163
  • 326 has no square factors that allow its square root to be simplified. √326 ≈ 18.055

Even though 326 is a three digit number, there is only one way to construct its factor tree, two if you count its mirror image. Either way is illustrated here. Below them is a Christmas factor tree puzzle that is a lot more interesting than the factor tree for 326.

2014-50 Level 4

Print the puzzles or type the factors on this excel file: 12 Factors 2014-12-15

2014-50 Level 4 Logic

321 I Heard the Bells on Christmas Day

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321 is made solely from three consecutive numbers so it can be evenly divided by 3.

  • 321 is a composite number.
  • Prime factorization: 321 = 3 x 107
  • The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1) = 2 x 2 = 4. Therefore 321 has exactly 4 factors.
  • Factors of 321: 1, 3, 107, 321
  • Factor pairs: 321 = 1 x 321 or 3 x 107
  • 321 has no square factors that allow its square root to be simplified. √321 ≈ 17.916
I heard the bells
Print the puzzles or type the factors on this excel file: 10 Factors 2014-12-08

One Christmas day about the time of the Civil War, Henry Wadsworth Longfellow wrote a poem he titled “Christmas Bells”. 5/7 of that poem later became the song known as “I Heard the Bells on Christmas Day”. You can read the poem’s complete original text here.

I heard the bells factors

315 Christmas Factor Trees

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315 is the smallest number that is divisible by every 1-digit odd number.

  • 315 is a composite number.
  • Prime factorization: 315 = 3 × 3 × 5 × 7, which can be written 315 = 3² × 5 × 7
  • The exponents in the prime factorization are 2, 1, and 1. Adding one to each and multiplying we get (2 + 1)(1 + 1)(1 + 1) = 3 x 2 x 2 = 12. Therefore 315 has exactly 12 factors.
  • Factors of 315: 1, 3, 5, 7, 9, 15, 21, 35, 45, 63, 105, 315
  • Factor pairs: 315 = 1 x 315, 3 x 105, 5 x 63, 7 x 45, 9 x 35, or 15 x 21
  • Taking the factor pair with the largest square number factor, we get √315 = (√9)(√35) = 3√35 ≈ 17.748

Five of the six factor pairs for 315 make lovely Christmas factor trees with the prime factors in red:

315 Christmas Factor Trees

Here’s a level 4 puzzle that is different kind of Christmas factor tree. Can you complete this mixed-up multiplication table when you’re given only these twelve clues?

2014-48 Level 4

Print the puzzles or type the factors on this excel file: 12 Factors 2014-12-01

2014-48 Level 4 Logic

306 and Level 4

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  • 306 is a composite number.
  • Prime factorization: 306 = 2 x 3 x 3 x 17, which can be written 306 = 2 x 3² x 17
  • The exponents in the prime factorization are 1, 2, and 1. Adding one to each and multiplying we get (1 + 1)(2 + 1)(1 + 1) = 2 x 3 x 2 = 12. Therefore 306 has exactly 12 factors.
  • Factors of 306: 1, 2, 3, 6, 9, 17, 18, 34, 51, 102, 153, 306
  • Factor pairs: 306 = 1 x 306, 2 x 153, 3 x 102, 6 x 51, 9 x 34, or 17 x 18
  • Taking the factor pair with the largest square number factor, we get √306 = (√9)(√34) = 3√34 ≈ 17.493

Happy Thanksgiving!

2014-47 Level 4

Print the puzzles or type the factors on this excel file: 10 Factors 2014-11-24

Here’s a little more about the number 306:

306 = 17 × 18, which means it is the sum of the first 17 even numbers.

  • Thus, 2 + 4 + 6 + 8 + . . .  + 30 + 32 + 34 = 306

It also means that we are halfway between 17² and 18², or halfway between 289 and 324. The average of those two numbers is 306.5.

AND it means that 17² + 18² – 1 = 2(306) = 2(17 × 18)

2014-47 Level 4 Logic

299 and Level 4

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  • 299 is a composite number.
  • Prime factorization: 299 = 13 x 23
  • The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1) = 2 x 2 = 4. Therefore 299 has exactly 4 factors.
  • Factors of 299: 1, 13, 23, 299
  • Factor pairs: 299 = 1 x 299 or 13 x 23
  • 299 has no square factors that allow its square root to be simplified. √299 ≈ 17.292

Warning: The logic needed to solve this Level 4 puzzle is complicated.

2014-46 Level 4

Print the puzzles or type the factors on this excel file: 12 Factors 2014-11-17

2014-46 Level 4 Logic

293 and Level 4

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  • 293 is a prime number.
  • Prime factorization: 293 is prime.
  • The exponent of prime number 293 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 293 has exactly 2 factors.
  • Factors of 293: 1, 293
  • Factor pairs: 293 = 1 x 293
  • 293 has no square factors that allow its square root to be simplified. √293 ≈ 17.117

How do we know that 293 is a prime number? If 293 were not a prime number, then it would be divisible by at least one prime number less than or equal to √293 ≈ 17.117. Since 293 cannot be divided evenly by 2, 3, 5, 7, 11, 13, or 17, we know that 293 is a prime number.

—————

I will be having major surgery tomorrow so I’ll publish the rest of this week’s puzzles early. I suspect that next week’s puzzles might be published late.

Today’s puzzle has another ten clues, but this puzzle is just a little more challenging that the last one.

2014-45 Level 4

Print the puzzles or type the factors on this excel file: 10 Factors 2014-11-10

Here are a few more number facts about the number 293:

17² + 2² = 293

Thus, 293 is the hypotenuse of a primitive Pythagorean triple:

  • 68-285-293 calculated from 2(17)(2), 17² – 2², 17² + 2².

293 is the short leg of this primitive Pythagorean triple:

  • 293-42924-42925

293 can also be written as the sum of three squares four different ways:

  • 16² + 6² + 1² = 293
  • 15² + 8² + 2² = 293
  • 14² + 9² + 4² = 293
  • 12² + 10² + 7² = 293

2014-45 Level 4 Logic

287 and Level 4

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  • 287 is a composite number.
  • Prime factorization: 287 = 7 x 41
  • The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1) = 2 x 2 = 4. Therefore 287 has 4 factors.
  • Factors of 287: 1, 7, 41, 287
  • Factor pairs: 287 = 1 x 287 or 7 x 41
  • 287 has no square factors that allow its square root to be simplified. √287 ≈ 16.941.

Here’s today’s puzzle:

2014-44 Level 4

 

Print the puzzles or type the factors on this excel file: 12 Factors 2014-11-03

2014-44 Level 4 Logic

280 and Level 4

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  • 280 is a composite number.
  • Prime factorization: 280 = 2 x 2 x 2 x 5 x 7, which can be written 280 = 2³ x 5 x 7
  • The exponents in the prime factorization are 3, 1, and 1. Adding one to each and multiplying we get (3 + 1)(1 + 1)(1 + 1) = 4 x 2 x 2 = 16. Therefore 280 has 16 factors.
  • Factors of 280: 1, 2, 4, 5, 7, 8, 10, 14, 20, 28, 35, 40, 56, 70, 140, 280
  • Factor pairs: 280 = 1 x 280, 2 x 140, 4 x 70, 5 x 56, 7 x 40, 8 x 35, 10 x 28, or 14 x 20
  • Taking the factor pair with the largest square number factor, we get √280 = (√4)(√70) = 2√70 ≈ 16.733

When I was a child, my brother showed me how he could multiply using his fingers. The same method is clearly illustrated here. If you haven’t seen it before, take a look. It isn’t the fastest method for multiplying, but it is still fun to see that it works. You can also practice the multiplication facts by finding the factors that solve this puzzle and then completing the multiplication table.

2014-43 Level 4

Print the puzzles or type the factors on this excel file: 10 Factors 2014-10-27

2014-43 Level 4 Logic